1 Introduction

Micro–nano structures with reduced period and increased inscribing area and productivity are being actively developed. Such structures show promise for applications in the integrated circuit industry as well as in the fields of physics [1, 2], materials science [3,4,5], chemistry [6, 7], and applications such as sensors [8,9,10,11,12], filters [13], and solar absorption [14,15,16]. Researchers are focusing on the fabrication of micro–nano structures [17,18,19], with various shapes and periods. Owing to the diffraction limit [20], the resolution of laser interference lithography [21, 22] can only reach half the wavelength of the incident light, making it difficult to satisfy the demand of device fabrication with high precision and integration density. Various micro–nano manufacturing techniques have been proposed to produce micro–nano structures with smaller periods, including extreme ultraviolet (EUV) lithography [23], which selects a shorter excitation wavelength, and immersion interference lithography [24, 25], which increases the numerical aperture of an optical system. These technologies afford high resolutions. However, the widespread application of EUV lithography has been hampered by the need for expensive equipment and complex processing. The mass-produced immersion interference lithography has been faced with the problem that the bubbles are difficult to be eliminated, and it may be that the lens and resist are in the environment of high refractive index liquid. Consequently, researchers are exploring alternative techniques. Recently, surface plasmon interference lithography [26, 27] and guided mode interference lithography [28, 29] have been widely used for the fabrication of high-resolution micro–nano structures.

Dong et al. [30] proposed a deep ultraviolet plasma structure based on grating coupling. For an excitation wavelength of 193 nm, a one-dimensional (1D) periodic grating can be obtained by irradiating the structure with transverse magnetic (TM) polarized light, and the two-dimensional (2D) periodic pattern can be obtained by irradiating a 2D grating with circularly polarized light. Gao et al. [27] experimentally demonstrated that a plasmonic cavity lens, which affords the advantages of scattering loss reduction with smooth silver films and evanescent waves amplified from the bottom silver layer, can improve the spread profile and resolution of surface plasmon lithography. Wang et al. [28] developed an asymmetric metal-cladding dielectric waveguide (AMCDW) structure based on zero-order mode interference lithography, which excited the existing TM0 mode. Sub-wavelength gratings with different periods and depth-to-width ratios can be obtained. However, in the aforementioned sub-wavelength lithography techniques, the thickness of the resist is generally of the order of tens of nanometers, making it difficult to fabricate a sub-wavelength grating under a thick resist and limiting practical lithography applications.

To solve these problems, this study proposes an interference lithography technique based on a symmetric metal-cladding dielectric waveguide (SMCDW) structure. This technique avoids the difficulties faced in fabricating the mask grating and realizes a sub-wavelength grating whose period can be adjusted by changing the resist thickness. Numerical results show that a sub-wavelength grating with a half-period of 46 nm (approximately \(\lambda /7\)) can be fabricated when a 325 nm laser is used to excite the TM0 mode in the SMCDW structure. The SMCDW structure can effectively avoid the thickness limitation of the resist, and a better sub-wavelength grating can be fabricated when the resist thickness is in the range of 80–120 nm. Moreover, the period of the sub-wavelength structure can be reduced.

2 Theoretical analysis

Figure 1 shows the schematic of interference lithography based on an SMCDW. First, a metal film layer is evaporated on the lower layer of an isosceles triangular prism and the upper layer of the substrate, and then, the resist is spin-coated on the substrate with the evaporated metal film to form a sample. The metal film at the lower layer of the prism is in close contact with the resist. In this manner, the resist is sandwiched between two Al film layers to form an SMCDW structure. In actual production, the matching oil [26, 29] with the same refractive index as the resist is often employed between the metal and resist, thereby avoiding the air gaps between them. Furthermore, in comparison with immersion interference lithography which uses a high refractive index liquid to improve the numerical aperture of an optical system, the proposed structure is intended to excite the TM0 mode in the waveguide structure by prism coupling to fabricate sub-wavelength grating. In immersion interference lithography, the interference pattern formed is obtained on the substrate, whereas the interference pattern formed here is etched on the metal film evaporated on the substrate. Al is selected as the metal for this structure with a 325 nm laser as the excitation source. The XOY plane is defined as the interface between the upper Al metal layer and the resist, with the Z direction being perpendicular to the interface and positive direction pointing to the prism.

Fig. 1
figure 1

Schematic of interference lithography based on the SMCDW structure comprising of a resist sandwiched between two metal film layers

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For TM polarized waves in the SMCDW structure, the in-plane wave vector, \({k}_{x}\), can be defined by the dispersion relation [31]

$$\varepsilon_{1} k_{z2} + \varepsilon_{2} k_{z1} {\text{coth}}\left( {\frac{{ - ik_{z1} d}}{2}} \right) = 0,$$
(1)

where \(k_{z}\) is defined by momentum conservation as

$$k_{z1,2}^{2} = \varepsilon_{1,2} \left( {\frac{\omega }{c}} \right)^{2} - k_{x}^{2} ,$$
(2)

where the subscripts 1 and 2 represent the resist layer and metal, respectively; \(\omega\) and \(c\) are the resonance frequency and the velocity of light in vacuum, respectively; and \(d\) is the thickness of the resist.

When two laser beams with the same amplitude irradiate an SMCDW at the same excitation angle, the TM0 mode can be excited and then propagated in the opposite direction, which will produce sub-wavelength optical field distribution. After exposure and development, a periodic sub-wavelength grating will be fabricated on the resist layer, and its period, \(\Delta x\), can be expressed as [28]

$$\Delta x = \frac{\lambda }{{2\sqrt {\varepsilon_{0} } {\text{sin}}\theta }},$$
(3)

where \(\varepsilon_{0}\) is the dielectric constant of the prism, \(\lambda\) and \(\theta\) represent the excitation light wavelength and excitation angle, respectively.

3 Results and discussion

3.1 Structural parameter selection

In the numerical simulation of TM0 mode interference in an SMCDW, the refractive indexes of the prism, resist, and substrate are 1.865, 1.53 [26], and 1.7, respectively; the optimized thicknesses of the upper and lower Al film are 15 nm and 25 nm, respectively; and the dielectric constant of the Al film under 325 nm laser is − 15.114 + 2.434i. Figure 2 shows the reflection spectra of the SMCDW structure when the resist thickness varies from 80 to 120 nm. Through the reflection spectra of these different resist thicknesses, it can be found that they all have narrow full width at half maximum, the minimum reflectivities are close to 0, and the corresponding excitation angles vary with resist thicknesses. Equation (3) indicates that the excitation angle is closely related to the period of the interference pattern. As the resist thickness increases, the excitation angle decreases and the period of the interference pattern increases, suggesting that the period of the sub-wavelength grating can be adjusted by changing the resist thickness. The thickness of the resist used here is 100 nm.

Fig. 2
figure 2

Reflective spectra of SMCDW structure when resist thickness ranges from 80 to 120 nm in steps of 10 nm

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3.2 Structure comparison

Next, we analyze the TM0 mode in the SMCDW structure in terms of the field intensity distribution, exposure depth, and other aspects. We also perform comparisons with an AMCDW structure. The AMCDW structure [28] is composed of an Al film, resist, and air; the materials selected are similar to those of the SMCDW structure. After optimizing the structure of AMCDW, the optimal thicknesses of the Al film and resist are estimated as 19 nm and 60 nm, respectively. As shown in Fig. 3, the excitation angles of SMCDW and AMCDW structures under optimal conditions are 70.80° and 55.49°, respectively. The SMCDW structure clearly has a larger excitation angle, implying that sub-wavelength grating fabricated has a smaller period.

Fig. 3
figure 3

Reflective spectra of SMCDW (red line) and AMCDW (blue line) structures. The thicknesses of the upper Al film, resist and lower Al film are 15, 100, and 25 nm in the SMCDW structure, respectively, whereas those of the Al film and resist are 19 and 60 nm in the AMCDW structure, respectively

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The intensity distribution in the cross section of the interference lithography structure is calculated by the finite element method, as shown in Fig. 4. When the laser is incident on a given structure at the corresponding excitation angle, resonance oscillation occurs with the free electrons in the upper Al film, causing a local field in the resist (between the second and third black dotted lines). In Fig. 4a, the optical field intensity in the resist layer is amplified longitudinally when it is close to the lower Al film, whereas in Fig. 4b, the optical field intensity decays rapidly in the resist layer and the intensity is almost 0 at the bottom of resist layer, even if the thickness of the resist selected in the AMCDW structure is thinner. Sub-wavelength gratings with half-periods of 46 nm (approximately \(\lambda /7\)) and 53 nm (\(\lambda /6.5\)) can be obtained after exposing and developing the resist, and the optical field distributions are shown in Fig. 4a and b, respectively. The sub-wavelength grating fabricated by TM0 mode interference in the SMCDW structure has a smaller half-period. Moreover, both positive and negative resists are suitable for the SMCDW structure, whereas only the negative resist can be used for the AMCDW structure.

Fig. 4
figure 4

Optical field intensity distribution of a SMCDW and b AMCDW structures. Black dotted lines represent the interfaces of different materials in the SMCDW and AMCDW structures

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In addition, because near-field imaging transmits sub-wavelength information through the field intensity, its imaging quality can be reflected by the exposure depth ratio. The relationship between the exposure depth ratio and different resist thicknesses in SMCDW and AMCDW structures is evaluated to further highlight the advantage of the former, as shown in Fig. 5. Among them, the inset figures show the field intensity distribution of SMCDW and AMCDW structures in the resist layer when resist thicknesses are 80 nm and 120 nm, respectively. The exposure depth ratio (\(\tau\)) is defined as the ratio of the exposure depth to the resist thickness. In Fig. 5, the exposure depth ratio of SMCDW is 1 as it is always parallel to the horizontal direction. This indicates that the exposure depth of the sub-wavelength grating fabricated from the SMCDW structure is consistent with the resist thickness, implying that the sub-wavelength information carried by the field intensity can be well transmitted to the other side of the resist and has better near-field imaging quality. By contrast, in the AMCDW structure, the exposure depth ratio decreases linearly with an increase in the resist depth. Therefore, the field intensity carrying sub-wavelength information cannot be effectively transmitted to the other side of the resist. Here, the exposure depth of AMCDW can be regarded as the distance at which the intensity amplitude decays to \(1/e\) of the surface value.

Fig.5
figure 5

Relationship between exposure depth ratio and resist thickness in the SMCDW (red line) and AMCDW (blue line) structures. Inset a and b are the optical field intensity distributions in the SMCDW structure when the resist is 80 and 120 nm, respectively; similarly, inset c and d correspond to the optical field intensity distribution in the AMCDW structure

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The aforementioned comparison shows that the SMCDW structure has the advantage of larger exposure depth. The optical field intensity and contrast of the interference pattern at different depths in the resist layer in the SMCDW structure are shown in Fig. 6, where the maximum and minimum values of interference pattern intensity are represented by \(\left| I \right|_{{{\text{max}}}}\) and \(\left| I \right|_{{{\text{min}}}} ,\) respectively. In the SMCDW structure, the total thickness of resist is 100 nm, and the resist depth is selected to start from 5 nm and increase to 95 nm in steps of 10 nm. It can be found that the maximum value of interference pattern intensity decreases first and then increases, whereas the minimum value is very small, almost 0. According to the definition of contrast, the contrast of resist at different depths is calculated, which is higher than 0.9. This shows that the sub-wavelength grating based on the SMCDW structure has the advantage of high contrast.

Fig. 6
figure 6

Optical field intensity (black line) and contrast (red line) of interference pattern at different depths of the resist layer when the resist thickness in the SMCDW structure is 100 nm

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4 Conclusion

In conclusion, this study proposes a method to fabricate sub-wavelength gratings based on the TM0 mode interference in an SMCDW structure. The numerical results show that a sub-wavelength grating with a half-period of 46 nm (approximately \(\lambda /7\)) can theoretically be obtained, and its period can be adjusted by changing the resist thickness. Irrespective of whether a positive or negative resist is used, better sub-wavelength gratings can be fabricated in practical lithography applications. The lithography method of the SMCDW structure has various advantages over asymmetric ones, including smaller periods of sub-wavelength gratings, higher contrast, and larger exposure depth of resist. This study offers a new approach for the development of near-field imaging and micro–nano lithography.